### Abstract

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, p_{c}, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to p_{c}. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to p_{c}. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

Original language | English (US) |
---|---|

Pages (from-to) | 57-69 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 137 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2009 |

### Fingerprint

### Keywords

- Continuum scaling limit
- Massive scaling
- Near-critical regime
- Percolation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*137*(1), 57-69. https://doi.org/10.1007/s10955-009-9841-y

**Trivial, critical and near-critical scaling limits of two-dimensional percolation.** / Camia, Federico; Joosten, Matthijs; Meester, Ronald.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 137, no. 1, pp. 57-69. https://doi.org/10.1007/s10955-009-9841-y

}

TY - JOUR

T1 - Trivial, critical and near-critical scaling limits of two-dimensional percolation

AU - Camia, Federico

AU - Joosten, Matthijs

AU - Meester, Ronald

PY - 2009/10/1

Y1 - 2009/10/1

N2 - It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, pc, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to pc. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to pc. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

AB - It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, pc, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to pc. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to pc. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

KW - Continuum scaling limit

KW - Massive scaling

KW - Near-critical regime

KW - Percolation

UR - http://www.scopus.com/inward/record.url?scp=70350622056&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350622056&partnerID=8YFLogxK

U2 - 10.1007/s10955-009-9841-y

DO - 10.1007/s10955-009-9841-y

M3 - Article

VL - 137

SP - 57

EP - 69

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -